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Condensation TechniquesBy Jürgen MüllerSee also: J. Müller, Computational representation theory. Remarks on condensation (dvi, ps, pdf), Lecture Notes, IEM, Universität DuisburgEssen, 2003. Condensation techniques are being used as one of the workhorses of computational representation theory, for both the analysis of given matrix or permutation representations and the explicit construction of matrix representations. Formally, these techniques are explicit computational applications of suitable Schur functors, see e. g. [Green]: Let A be a finitedimensional algebra over a field F, and let e be an idempotent in A. Then the corresponding Schur functor maps an Amodule V to the eAemodule Ve, where the subset Ve of V is the image of the projection induced by e.
In the group algebra case A=FG, where G is a finite group, this is
applied as follows: Let H be a subgroup of G such that the
characteristic of F does not divide the order of H. Then
e = H^{−1} ∑_{h ∈ H} h is an idempotent in FG. The subset Ve of an FGmodule V is the set of Hfixed points in V, hence this technique is called fixed point condensation. Algorithms for fixed point condensation have been developed, implemented and used by various people, see e. g. [Müller] for a more detailed overview. In particular, the following applications underline the role of GAP as a valuable research tool: Finding socalled faithful idempotents has been pursued in [Lux], where in particular the character tables and tables of marks libraries of GAP have been used. Fixed point condensation of induced modules has been described and implemented as GAPcode in [MüllerRosenboom]. Socalled direct condensation of permutation modules is based on the general computational task of enumerating large Gsets. Programs using the technique of distributed computing have been described in [LübeckNeunhöffer], where the overall implementation is as Ccode, but certain precomputation programs are written as GAPcode. An application of direct condensation techniques is given in [MüllerNeunhöfferRöhrWilson], where again both Cprograms and GAPprograms are used. Enumeration of huge Gsets, together with applications, has been described in [Müller], where the latter programs are written completely as GAPcode. Typically, the GAPparts of the abovementioned implementations make use of the basic data structures and arithmetical features of GAP, such as permutations or vectors and matrices over small finite fields, and the possibility to handle them efficiently and manipulate them easily. These specially tailored programs are not part of the official GAPdistribution, and for more details the reader is referred to the various authors. References
[Green]
J. Green: Polynomial representations of GL_{n},
Lecture Notes in Mathematics 830, Springer, 1980.
